There exists a difference of opinions between Vadakalai and Tenkalai traditions – like, differences in the religious practices of the Achary-s, differences in doctrine, etc. It is enough that one recites the thaniyans of their respective Acharya-s. It is not a mistake to recite other thaniyans during special occasions.
Notes:
Each thaniyan glorifies that Acharya who is mentioned by the thaniyan and not who composed or recited that thaniyan. It is better to recite thaniyan of one’s respective Acharya. As we belong to Desika Sampradaya, it is enough that we recite Desikan thaniyan.